Subject: Re: two questions
From: Jeffrey Strom
Date: Thu, 22 Jun 2006 22:23:25 -0400
>
> Subject: weak equivalence pi_k(X)-->pi_k(Y) if k>n
> From: Gaucher Philippe
> Date: Thu, 22 Jun 2006 16:00:01 +0200
>
> Dear All
>
> Is there a known construction on the category of compactly generated
> topological spaces, or on the category of simplicial sets of a model
category
> structure such that X-->Y is a weak equivalence iff pi_k(X)-->pi_k (Y)
is an
> isomorphism for k > n (n fixed).
>
> (for k S^n-->D^{n+1}).
>
> Thanks in advance. pg.
Just a thought about this.
I think Neisendorfer localization is relevant here (that is,
nullification with respect to BZ/p followed by p-completion --
I'll denote it by L).
If X is a 2-connected finite complex which is p-complete
(i.e., finite, p-local homotopy groups), and X(n) is its
n-connected cover, then L(X(n)) = X, whatever n is.
Therefore, if f:X --> Y induces isomorphisms pi_k(X) --> pi_k(Y)
for k > n, then X(n) --> Y(n) is a weak equivalence,
and (assuming the Neisendorfer result is natural enough,
which I'm pretty sure it is), then
X = L(X(n)) --> L(Y(n)) = Y
is also a weak equivalence.
Jeff